Let $M$ be a compact Riemannian manifold and $TM$ be its tangent bundle. Given a initial point-vector $(x,v) \in TM$ and a curve $\alpha:[0,1] \to M$ starting at $x$ we can parallel transport $(x,v)$ along $\alpha$ to obtain a point vector $(y,w)$.
The most natural question is which vectors can be connected in this way. Obviously the norm of the two vectors must be the same.
One situation is that there is a smaller dimensional subbundle of $TM$ which is preserved by parallel transport. In this case de Rhams theorem says that $M = K\times L$ for some Riemannian manifolds $K$ and $L$, so this case is uninteresting.
In the case where there is no invariant subbundle of $TM$, Berger's theorem states that either $M$ is a locally symmetric space with rank 2 or more or one can take any vector to any other of the same norm. In the second case we say holonomy is transitive on $M$ (or more precisely on the unit tangent bundle $SM$).
I'd like to consider the orthogonal frame bundle $OM$. This is a connected manifold if $M$ is non-orientable and has two components if $M$ is orientable.
My question is the following: If holonomy is transitive on the unit tangent bundle $SM$ is it necesarilly transitive on each component of the orthogonal frame bundle $OM$?
Thanks for the help!